In Part I of this paper, an abstract analogue of the minimization problem for Boolean functions and of the notion of prime implicant is defined, so that this general problem can be solved in the same steps as in the classical case: 1) determination of the prime implicants; 2) determination of all the solutions made up of prime implicants. In Part II it is shown that the classical minimization problem, as well as certain set-theoretical and graphtheoretical problems are particular cases (...) of the general problem defined in Part I. (shrink)

W pierwszej części tej pracy określa się pewien abstrakcyjny analogon problemu minimalizacji dla funkcji Boole’owskich oraz pojęcia implikantu pierwszego, w taki sposób, że ten ogólny problem może być rozwiązany za pomocą tych samych kroków co w przypadku klasycznym, tzn. przez: 1) znalezienie implikantów pierwszych, 2) znalezienie wszystkich rozwiązań utworzonych z implikantów pierwszych. W drugiej części pokazuje się, że klasyczny problem minimalizacji oraz pewne problemy teorio-mnogościowe a nawet pewne problemy z zakresu teorii grafów są szczególnymi przypadkami ogólnego problemu sformułowanego w części (...) pierwszej. (shrink)

We study ranges of algebraic functions in lattices and in algebras, such as Łukasiewicz-Moisil algebras which are obtained by extending standard lattice signatures with unary operations.We characterize algebraic functions in such lattices having intervals as their ranges and we show that in Artinian or Noetherian lattices the requirement that every algebraic function has an interval as its range implies the distributivity of the lattice.

We study ranges of algebraic functions in lattices and in algebras, such as Łukasiewicz-Moisil algebras which are obtained by extending standard lattice signatures with unary operations.We characterize algebraic functions in such lattices having intervals as their ranges and we show that in Artinian or Noetherian lattices the requirement that every algebraic function has an interval as its range implies the distributivity of the lattice.

The set (X, J) of fuzzy subsetsf:XJ of a setX can be equipped with a structure of -valued ukasiewicz-Moisil algebra, where is the order type of the totally ordered setJ. Conversely, every ukasiewicz-Moisil algebra — and in particular every Post algebra — is isomorphic to a subalgebra of an algebra of the form (X, J), whereJ has an order type . The first result of this paper is a characterization of those -valued ukasiewicz-Moisil algebras which are isomorphic to an algebra (...) of the form (X, J) (Theorem 1). Then we prove that (X, J) is a Post algebra if and only if the setJ is dually well-ordered (Theorem 2) and we give a characterization of those -valued Post algebras with are isomorphic to an algebra of the form (X, J) (Theorem 3 and Proposition 2). (shrink)